A cubic nanometer volume
of a material with a (typical) volumetric heat capacity
of 10 6 J/m 3ĚK has a heat
capacity of 1 maJ/K.

Thermal conductance scales like
electrical conductance, with

thermal conductance L

(2.41)

and a cubic nanometer of
material with a (fairly typical) thermal conductivity of
10 W/mĚK has a thermal conductance of 108
W/K.

Characteristic times for thermal
equilibration follow from these relationships, yielding

thermal time constant L2

(2.42)

For a cubic nanometer
block separated from a heat sink by a thermal path with a
conductance of 108 W/K, the calculated
thermal time constant is ~ 1013 s, which
is comparable to the acoustic transit time. (In an
insulator, a calculated thermal time constant approaching
the acoustic transit time signals a breakdown of the
diffusive model for transport of thermal energy and the
need for a model accounting for ballistic transport; in
the fully ballistic regime, time constants scale in
proportion to L, and thermal energy moves at the
speed of sound.)

The scaling relationship for frictional
power dissipation, Eq. (2.16), implies a scaling law for
the temperature elevation of a device in thermal contact
with its environment,

temperature elevation L

(2.43)

This indicates that
nanomechanical systems are more nearly isothermal than
analogous systems of macroscopic size.